by the standard upper bound for the supnorm of an eigenfunction on finite volume symmetric spaces [1]},
\(\mathrm {Vol}(3\Lambda )\sum _{0\ne \mu \in \lambda ^{-3} \Lambda } |\rho _j(\mu )|^2 y^2 K_{2it_j} (4\pi |\mu | y)^2=\iint _{\mathbb {C}/3\Lambda } |\phi _j|^2 dx_1dx_2 \le \sup |\phi _j|^2 \ll |t_j|^2\) 
Assume without loss of generality that \(t_j>100\) , and, for a given \(\mu \in \lambda ^{-3}\Lambda \) ,
choose \(y\)  so that \(4\pi |\mu | y = 2t_j\) . Then we have
\(|\rho _j(\mu )|^2 \frac{t_j^2}{4\pi ^2|\mu |^2} K_{2it_j} (2t_j)^2 \ll t_j^2,\) 
and so using the asymptotic of the K-Bessel function in the transition range [2]},
\(|\rho _j(\mu )| \ll t_j^{\frac{1}{3}} e^{\pi t_j} |\mu |.\) 
